Real methods in complex analysis

7/23/2019 Real methods in complex analysis

1/23

REAL

PROOFS

OF

COMPLEX THEOREMS (AND

VICE VERSA)

LAWRENCE

ZALCMAN

Introduction.

It has become

fashionable

recently

to

argue

that

real and

complex

variables should be taught together as a unified curriculumin analysis. Now this is

hardly

a novel idea,

as

a

quick

perusal

of Whittaker

and Watson's

Course

of

Modern

Analysis

or either Littlewood's

or Titchmarsh's

Theory

of Functions (not

to

mention

any

number

of cours

d'analyse

of the nineteenth

or twentieth century)

will

indicate.

And, while

some

persuasive

arguments

can be advanced

in favor of this

approach,

it

is by

no

means

obvious

that the advantages

outweigh the disadvantages

or,

for

that matter,

that a unified treatment

offers

any

substantial

benefit to

the

student.

What

is obvious

is

that the two

subjects

do

interact,

and interact

substantially,

often

in

a

surprising

fashion. These points

of tangency present

an instructor

the

opportunity

to

pose

(and

answer)

natural

and

important

questions

on basic

material

by

applying

real

analysis

to

complex

function

theory,

and vice versa. This

article

is

devoted

to

several

such

applications.

My

own

experience

in

teaching suggests

that the

subject

matter discussed

below

is

particularly

well-suited

for

presentation

in a

year-long

first

graduate

course

in

complex analysis.

While

most of this material

is

(perhaps

by definition)

well

known

to the

experts,

it is

not,

unfortunately,

a

part

of

the

common

culture of

professional

mathematicians.

In

fact,

several

of

the

examples

arose

in

response

to

questions

from friends and colleagues. The mathematics involved is too pretty to be the private

preserve

of

specialists.

Publicizing it is

the

purpose

of the present

paper.

1. The Greening

of

Morera. One

of the

most

useful

theorems

of basic complex

analysis

is the

following result,

first

noted

by

Giacinto Morera.

MORERA'S

THEOREM

37].

Let

f(z)

be a continuous

function

on the domain

D.

Suppose

that

(1)

f(z)dz

=

0

for every

rectifiable

closed

curve

y lying

in

D.

Then f is holomorphic

in D.

Morera's

Theorem enables

one

to

establish the

analyticity

of functions

in

situations

where resort to the

definition

and

the attendant calculation of

difference

quotients

would

lead to

hopeless complications.

Applications

of this sort

occur,

for

instance,

in the

proofs

of the

Schwarz Reflection Principle and other

theorems

on the

extension

of

analytic

functions.

Nor

is its usefulness

limited

to this

circle of ideas;

the

important

fact

that

the uniform limit of

analytic

functions

is

again

analytic

is an

immediate consequence (observed already by Morera himself, as well as by Osgood

[39],

who had

rediscovered

Morera's

theorem).

115


2/23

116

LAWRENCE

ZALCMAN

[February

Perhaps

surprisingly,

the

proofs

of Morera's theorem found

in complex analysis

texts

all follow a single pattern.

The hypothesis

on

f

insures

the existence of a

single-valued

primitive

F of

f,

defined

by

rz

(2)

F(z)

f(C)dC.

0

Here

zo

is some

fixed

point

in

D

and

the

integral

is taken over

any

rectifiable

curve

joining

zo

to z. The function

F is

easily

seen to be

holomorphic

in

D,

with

F'(z)

=

f(z);

since the

derivative

of a

holomorphic

function is

again

holomorphic,

we

are done.

Several remarks

are in order concerning

the proof

sketched above.

First

of all,

the assumption

that

(1)

holds

for all rectifiable closed

curves

in D is much too

strong.

It

is

enough,

for

instance,

to assume

that

(1)

holds for all

closed curves

consisting

of a finite number of straight line segments parallel to the coordinate axes; the

integration

in

(2)

is then effected

over a

(nonclosed)

curve

composed

of such

segments,

and the proof proceeds

much

as before.

Second,

since

analyticity

is a

local

property,

condition

(1)

need

hold

only

for an

arbitrary neighborhood

of each

point

of

D;

that

is,

(1)

need

hold

only

for small curves.

Finally,

the

proof requires

the

fact

that

the derivative

of

an

analytic

function is

again

analytic.

While this

is a

trivial con-

sequence

of the

Cauchy integral

formula,

it can be

argued

that

that

is an

inappropriate

tool for

the problem

at

hand;

on the other

hand,

a

proof

of this fact

without

complex

integration

is

genuinely

difficult

and

was,

in

fact,

only

discovered

(after many years

of

effort)

in

1961

[44],

[10], [46].

There is an additional

defect

to the

proof,

and that is that

it

does not

generalize.

Thus, it was more

than

thirty years

after

Morera

discovered his

theorem that

Torsten

Carleman

realized

the

result

remains

valid

if

(1)

is

assumed

to

hold

only

for

all

(small) circles

in D.

It is

an

extremely

instructive

exercise

to

try

to

prove

Carleman's

version

of Morera's

theorem

by mimicking

the

proof given

above. The

argument

fails

because it cannot

even

be started:

the

very

existence of

a

single-valued

primitive

is in

doubt.

This

leads

one to

try

a different

(and

more

fruitful) approach,

which

avoids the use of primitives altogether.

Suppose

for

the

moment

that

f

is a smooth

function, say

continuously

dif-

ferentiable.

Fix

zo

e

D and

suppose

(1)

holds

for

the circle

Fr(ZO)

of radius

r,

centered

at

zo.

Then,

by

the

complex

form of

Green's.

theorem

0 f(z)dz

=

2i

jj

dxdy,

Z O )

Z O o f

where

4(Zo)

is

the

disc

bounded

by

FU(zO)

and

Of/lz

=

2

(3f/Ox

+

i

Of/Dy).

(There's

no cause

for

panic

if the

0/Di operator

makes

you

uneasy

or

you

are not

familiar

with the

complex

form of Green's

theorem; just

write

f(z)

=

u(z) + iv(z),

dz

=

dx

+ idy,

and

apply

the

usual version

of Green's theorem

to the real and

imaginary

parts

of

the

integral

on

the

left.) Dividing by

an

appropriate

factor,

we


3/23

1974]

REAL

PROOFS OF COMPLEX THEOREMS

117

have

7tr2 t

(:) jedxdy =

0;

i.e., the average of the continuous function Of/0 over the disc

AF(zo)

equals 0.

Make

r

-+

0 to obtain (Of

O)

(z0)

=

0.

Since

this holds

at

each

point

zo

e

D,

Of

Oz=

0

identically

in

D. Writing this

in real

coordinates,

we see

that

ux

=

vp,

uY

=

-vx

in D; thus the Cauchy-Riemann equations

are

satisfied

and

f

is

analytic.

Notice that we did not need to assume

that

(1)

holds

for

all

circles in

D or

even

all small

circles;

to

pass

to

the

limit it was

enough

to

have,

for each

point

of

D, a

sequence of circles shrinking to that point. Moreover, since f

has

been assumed

to

be continuously differentiable, it is sufficient to prove that

af/iz

vanishes on

a

dense set. Finally, and most important, the fact

that our curves were circles

was

not used

at all Squares, rectangles, pentagons,

ovals

could have been used

just

as well. To conclude that

(Of/li)(zo)

=

0, all

we

require

is that

(1)

should

hold

for a

sequence

of

simple closed

curves

y

that accumulate

to

zo

(zo

need

not

even

lie

inside or on the y's) and that the curves involved

allow

application

of

Green's

theorem. It is enough, for instance, to assume that the curves are piecewise con-

tinuously differentiable.

To summarize, we have shown that Green's theorem yields in a simple fashion

a

very general and particularly appealing version

of

Morera's

theorem for C'

func-

tions. It may reasonably be asked at this point if the proof of Morera's theorem

given above can be modified to work for functions

which

are

assumed

only

to

be

continuous. That is the subject of the next section.

2.

Smoothing. Let +(z) be a real valued function defined

on

the

entire

complex

plane which satisfies

(a)

0(z)

>

09

(b)

ff

q(z)dxdy

=

1,

(c)

4

is

continuously differentiable,

(d) 0(z) =

0

for Iz I _ 1.

It is trivial to construct such functions; we can even require

/

to be infinitely dif-

ferentiable and to depend only on

I

z

,

but these properties will not be required in the

sequel. Set, for

e

> 0,

4,(Z)

=

C-20(z/E).

Then, clearly, 4? satisfies (a) through (c)

above

and

/E

vanishes off

j

I

O

>

n

2

so that f

is

univalent.

The

second

ingredient we

need is the celebrated Hadamard

gap

theorem.

HADAMARD

GAP THEOREM.

Let f(z)

=

U=O

a

znk

have A

as its

disc

of con-

vergence. If

nk+link

_

qfor

some q > 1 and all

large k,

thenf

has

F

as

its

natural

boundary; that is, f cannot be continued analytically across any subarc of F.

The

beautiful proof

of this theorem due to L. J. Mordell

([36],

cf.

[54,

p. 223])

should be

standard fare in

graduate

courses in complex

analysis.

The

construction of

the

required function is

now almost trivial.

We

choose the

sequences

{ak}

and {nk}

to satisfy

(a)

ao

=

no

=

1,

(b)

zko1

nkf

ak| 0 [61]. Even this condition is not necessary;

in

fact,

Denjoy [11]

has constructed an arc which

is

the graph of a function and

which

has the

required property.

6.

Blowing up

the

boundary. Questions involving length and area arise in con-

formal

mapping

as well.

A

conformal

map, being

analytic,

must map sets of

zero

area to

sets

of zero area; however, distortion at the boundary is an a priori possibility.

Writing

A

u

F

=

{z:

z

f

?

1}

as

before,

let us assume

that

the

univalent function

f(z) maps

A

conformally

onto

the Jordan region D. According to the Osgood-

Taylor-Caratheodory theorem,

f

extends to a homeomorphism of A u F onto

D

u OD.

(Proofs

of

this

important result, announced by Osgood [65] and proved

independently by Osgood

and

Taylor [66, p. 294], and

Caratheodory

[6], [7] are

available

in

[9, pp. 46-49] and [24, p. 129-134]. The reader will find a comparison

of the

treatments

in

these

references particularly instructive in the matters of style

of

exposition

and

attention

to

detail.) In case OD is rectifiable, a theorem of the

Riesz brothers

[47]

insures that

f

and

f

-'

preserve sets of zero length (

=

Haus-

dorff

one-dimensional

measure). When

OD

fails to be

rectifiable,

however, all hell

breaks loose.

In

particular,

a

subset of

OD

having positive area may correspond to

a subset of F having zero Lebesgue (linear) measure For the construction, we need

an

important result from plane topology.

MOORE-KLINE

EMBEDDING THEOREM

[351.A necessary and sufficient condition


10/23

124

LAWRENCE ALCMAN

[February

that

a

compact

set K

(

C

should

lie

on

a

simple

Jordan arc is that

each

closed

connected subset

of

K

should be either a

point

or

a

simple

Jordan arc with

the

property that

K

-

y

does

not

accumulate at

any point of

7,

except

(perhaps) the

endpoints.

Now

let K be

a

Cantor set

having positive

area;

K

may

be

realized,

for

instance

as the product of

two linear Cantor

sets,

each

of which has

positive

linear

measure.

Construct

countably

many disjoint

simple

Jordan arcs

Jn

c

C

\ K

such that

the

sequence

{Jn}

accumulates

at

each

point

of K and at no other

points

of

C and

with

the

additional

property that

if

zo

e

C\

(K

tu

{Jn})

=

R and

z e

K,

then

any

arc

from

zo

to

z

which lies,

except

for its final

endpoint,

in

R

must

have

infinite

length.

By

the Moore-Kline

embedding

theorem,

we

may pass

a

simple

closed Jordan

curve J through K

U

{JnJ.

Let

f

be a conformal

map

from

A to

D,

the

domain

bounded

by

J. Then

f extends to a

homeomorphism

from

F

to J.

Let

S

=

f

-'(K).

That

S has zero linear

measure

follows at once from the

following

theorem,

due

to

Lavrentiev.

THEOREM.Let

f be a

conformal

homeomorphism

of

Au

Fu

onto

the Jordan

domain D

UJ.

If

S

c

J is

not

rectifiably

accessible

from

D

then

f

'(S)

c

F

has

zero

measure.

Proof.

Since

D is a

bounded domain, its

area,

given by the

expression

f fA

f '(z)

I2dxdy,s

finite. Thus

2n

1

f

f'(reio)

I

rdrdO

2ir 1

1

2

2r2

1

112

?

(Jj

k O

rdrdO

If'(reE?) r

drdO

0. The

first

person to

show that

a set of

capacity zero

on

F

could correspond

under a

conformal

mapping to a set having positive area was Kikuji Matsumoto [33]. He actually

proved

(what is implicit in

the

above discussion)

that for each

totally

disconnected

compact subset K

of the plane

there

exists a Jordan

domain D with

boundary

J

D

K

such

that K

corresponds

under

conformal

mapping to a set of

capacity

zero


11/23

1974]

REAL PROOFS OF COMPLEX THEOREMS

125

on

F. The discussion here

(in

particular,

the ingenious proof of the central result)

is based

on an idea of Walter Schneider

[51].

The compression

of

the

boundary

of

the unit disc

presents

greater difficulties.

Lavrentiev, however, has shown that a set of positive measure on F may be mapped

onto a set of zero length under a conformal mapping of Jordan

domains

[30].

A

more recent construction is due to McMillan and Piranian

[32].

7. Absolute convergenceand uniformconvergence.

Conformal

mapping

techniques

are also useful in constructing examples concerning the convergence of

power

series and Fourier series. Below we offer some simple but instructive

examples.

The first example

of a

power

series which

converges

uniformly

but not

absolutely

on

the closed unit disc was

given by Fejer [15],

cf.

[25,

vol.

1,

p. 122].

The

following

geometric example, due to Gaier [68] and rediscovered by Piranian (see [1, pp. 289,

314]),

is

particularly appealing.

Let D be the region of figure 1, a triangle from

FIG.

1

which

wedges have been removed

in such a way that the vertex at z

=

1

is not

rectifiably accessible from the

interior of D. Since D is a Jordan

region, any conformal

map

of

A

=

{z:

z

(x)

e

Y, q$(xk)

e

J

(k

=

1, 2,

)

and

5

is

closed

under

linear

combina-

tions. Since

(by

hypothesis) x

e

5, each

polynomial vanishing

at the origin

belongs

to

S.

The proof

depends on a

simple lemma

concerning the

approximation

of

functions.

LEMMA.

Let

0(x)

satisfy

(a). Suppose that

for

each

E

>

0

there

exist

poly-

nomials

pl(x),

p2(X) such that

pi(O)

=

0,

pi(l)

=

1 (i

=

1,2) and

P(x)

N. (The

inner circle

contracts,

the

outer circle

expands, and

the notch

gets

thinner and rotates,

tending

toward

but

never

reaching its

limiting line.)

The

function

0? z cKn

gn(Z)

{ K

clearly

extends to

a function

analytic

in a

neighborhood

of the (disconnected)

set

Kn

u

{0}.

Since

this set does

not separate the

plane,

g may be

approximated

uni-

formly (to

within

1/n, say) on

Kn

u

{0}

by

a polynomial

pn.

These polynomials

clearly

satisfy

(11).

12. Miscellany.

The interactions

between real

and complex

analysis

are by no

means limited to the areas mentioned above. To keep the discussion within manage-

able

limits,

we have restricted

ourselves

to

(a

subset of)

those

applications,

examples,

and

aspects

of the theory

that have

not

found sustained

treatment

in the "popular"

literature of texts

and survey

articles.

Subjects which

are

treated elsewhere

at ade-

quate

length but

which deserve

mention

here by

virtue of their

interdisciplinary

nature include

the following:

(a) The evaluation

of real

integrals

and sums

by

residue techniques.

This is

surely

one of the

most striking

applications

of complex

function

theory to real

analysis. Fortunately,

any

good text on

complex

analysis will

contain a

fairly detailed

discussion.

(b)

Complex methods

in harmonic

analysis.

This is a substantial area,

which

includes topics

as

diverseas interpolation

theorems

(see,

for instance,

[28, pp. 93-98])

and

theorems

of Paley-Wiener

type [43].

Two

of the most

attractive recent

texts

in

harmonic analysis

[13], [28]

devote

whole chapters

to this

aspect

of the theory.

Further

developments are

discussed

in

the survey article

of

Weiss [57].

(c)

Functional

analysis. Complex

variable

methods

appear

here

perhaps

most

notably in the construction

of

functional

calculi

for operators

on Hilbert

space

or

Banach space. The applications to commutative Banach algebras are particularly

substantial;

indeed, parts

of this

last-named

subject

are

virtually

coextensive

with

certain

aspects

of

several

complex

variable

theory.

For

further

references,

see

[17]

and

[58].

In the

opposite

direction,

techniques

of

functional

analysis

can be

used


20/23

134 LAWRENCE ZALCMAN

[February

to

establish many results in

function theory; this is the programme of [48].

Finally,

an

honest partnership between

complex variables and functional analysis occurs

in

the

study of certain Banach spaces of analytic functions, especially HP

spaces

[12],

[26].

(d) Function theoretic methods in

differential

equations. Complex

methods

occur rather naturally in the

study of ordinary differential equations [8].

Their

appearance in the study of

partial differential equations is perhaps more

surprising.

Yet there are substantial applications, and more than one book [2], [19]

has been

devoted to this area. Further applications of function theory to

problems in partial

differential equations will be

found in [18]. In a rather different direction, the theory

of

linear partial differential equations with constant coefficients is

intimately con-

nected with the study of certain spaces of entire functions of several

complex varia-

bles; see [14] for an exhaustive treatment.

13. Monodromy. No excursion onto the bypaths of complex analysis would be

complete without some

mention of the monodromy theorem.

MONODROMY THEOREM.

Let D be a simply connected domain and let

f(z)

be

analytic in a neighborhood of

zo

e

D. Then if f(z) can be continued

analytically

from

zo

along every path lying in D, the continuation

gives]

rise to a

single-valued

unction analytic on all

oJ

D.

A more general version states that analytic continuation along paths is a homo-

topy invariant; see, for

instance, [53]. Like the reflection principle, the monodromy

theorem is an essential

ingredient in the short proof of Picard's little theorem;

in

its

extended

form, it is the

central result in the subject of analytic continuation.

Yet

no

theorem of basic complex analysis is more abused or less understood.

Indeed,

it

has been

misapplied more than once even by mathematicians of

the

first

rank

(and

specialists in complex analysis,

at that ). One may speculate that

a source

of at

least

some of the confusion surrounding this result is the essentially

topological,

rather than function-theoretic,

nature of the theorem.

The sort of error into which one may lapse is best indicated by an explicit example.

Let

D

be a simply connected

domain and f a function analytic

in D which satisfies

f '(z)

0

0 on D.

Suppose R

=

f(D)

is

also

simply

connected.

Question:

Must

f

be

univalent

(one-one)? An affirmativeanswer may be found in [56, p. 243] and

in

other

references as well. The

argument is as follows. At each point w0

E

R one may

define

a

local inverse fJt1(w) of f,

analytic in a neighborhood of w0. Since R

is

simply

connected, the totality of these functions defines a single-valued analytic

function

f

-1

on

R, which is a global

inverse for f. Thus f must be univalent.

Note

further

that

the

simple-connectivity of

D

is quite extraneous to the demonstration.

Unfortunately,

the

argument given above

is

altogether incorrect,

since

the

essential hypothesis of the

monodromy theorem, that analytic continuation

be

possible along every path

in

R, has not been verified.

Can

the proof

be

salvaged?


21/23

1974]

REAL

PROOFS

OF

COMPLEX

THEOREMS

135

The answer

is no. In fact,

consider the

function f(z)

=

f

e2

dC.

This

f

is analytic

(entire) on

the simply

connected domain C

and f'(z) _

eZ2

is

nowhere

zero. Clearly,

f is

not univalent. So

it suffices to prove

that

f(C)

is simply

connected. We claim

f(C)

=

C. Indeed, suppose f(z) # w. Since

eZ2

is an even function, f(z) is odd,

so

that

if

f

fails to take on the

value

w

it

also misses the value

-

w.

If w

=A

,

this

contradicts

Picard's

(small) theorem.

Since

f(O)

=

0,

f takes on

every

value in

the

complex plane.

For

D

=

C, any

function

which satisfies f'(z)

0

0 must be

transcendental

and hence must (by

Picard's

theorem) take on most values

infinitely

often.

One

can,

however, construct a

non-univalent, locally univalent function

mapping

the

disc

A

=

{z:

I

z

I

Real methods in complex analysis

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Transcript of Real methods in complex analysis